My conclusion are --
1. Based on visual evidence of what QEQM wore for various trips to parts of Wales in 1937, 1945 and 1957, I concluded that the brooch she wore was part of the gift of the Ladies of North Wales to Queen Alexandra.
For some of this material see https://mb.boardhost.com/historyroyaljewels/msg/1551950241.html
The UK press has constantly confused the brooch from North Wales with that from South Wales. It is possible that the confusion has arisen because of press releases from the Palace, or that newspapers have copied a report put out by a syndicated news service without checking details.
Ladies of North Wales gift.
This is the brooch which Queen Elizabeth the Queen Mother was photographed wearing on her trips to Wales.
For other images see the link I posted above. For videos please see this link
The gift from South Wales was/is a large brooch pendant. There does not appear to be any evidence that Queen Elizabeth the Queen Mother wore it on her trips to Wales. We do know that, after she became Queen, she frequently wore the bracelet which was part of the gift from South Wales. But it was never seen on her prior to 1937.
Brooch/pendant from South Wales
Bracelet from South Wales.
Re what was displayed in 1957
I have never been able to find an image of the brooch which was displayed during (and after) the charity ball held in 1957.
There are only two descriptive reports which include details about the brooch displayed, and one advertisement.
There are discrepancies in the reports.
For example the gift from the Ladies of North Wales was given in June 1863.
The gift from South Wales was presented On St David's Day, 1 March 1864.
The gift of the Ladies of North Wales was presented in June 1863.
The press reports about the gift lent for display at the charity ball in 1957 talk about the leek being depicted by a pearl, which would fit the description of the jewel from South Wales.
Yet, there is no visual evidence (or descriptive evidence) which I have come across showing that Queen Elizabeth the Queen Mother ever wore this brooch to functions either in Wales or in connection with Wales.
The evidence is that she wore the large brooch which was the present from North Wales.
Perhaps the elaborate (almost unwearable gift pendant/brooch) from South Wales was the one displayed in 1957. But there is no evidence it was ever worn by QEQM during her public appearances in Wales.
Was the jewel left to George VI?
For a number of reasons, I doubt it. As we all know Queen Alexandra died intestate. Neither the brooch or the bracelet from South Wales or the large brooch from North Wales is recorded as having been worn publicly by QEQM. before she became Queen Consort. We do know that she wore the emerald earrings which were part of the gift of North Wales from shortly after the time of Queen Alexandra's death. Only after she became Queen were the bracelet from South Wales and the brooch from North Wales seen in photographs. If they had been "left" to the then Duke of York there seems no reason why she would not have worn them. (We are talking about an 11 year gap between Queen Alexandra's death and when QEQM became Queen Consort)
I suspect that the gifts from South Wales and the large brooch from North Wales were kept by George V and Queen Mary. If Queen Alexandra, who died intestate, indicated who was to receive these jewels in her jewel inventory (as noted by Hugh Roberts), I suspect that the recipient would have been the then Prince of Wales, later Edward VIII.